Arbitrarily High-Order Time-Stepping Schemes for Nonlinear Klein–Gordon Equations

Abstract

This chapter presents arbitrarily high-order time-stepping schemes for solving high-dimensional nonlinear Klein–Gordon equations with different boundary conditions. We first formulate an abstract ordinary differential equation (ODE) on a suitable infinite–dimensional function space based on the operator spectrum theory. We then introduce an operator-variation-of-constants formula for the nonlinear abstract ODE. The nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix. With regard to the two dimensional Dirichlet or Neumann boundary problems, the time-stepping schemes coupled with discrete Fast Sine/Cosine Transformation can be applied to simulate the two-dimensional nonlinear Klein–Gordon equations effectively. The numerical results demonstrate the advantage of the schemes in comparison with the existing numerical methods for solving nonlinear Klein–Gordon equations in the literature.